CSE 321 Discrete Structures

Winter 2008Lecture 20

Probability: Bernoulli, Random Variables, Bayes’ Theorem

Announcements

• Readings – Probability Theory

• 6.1, 6.2 (5.1, 5.2) Probability Theory• 6.3 (New material!) Bayes’ Theorem• 6.4 (5.3) Expectation

– Advanced Counting Techniques – Ch 7.• Not covered

– Relations• Chapter 8 (Chapter 7)

Highlights from Lecture 19

• Conditional Probability• Independence

Let E and F be events with p(F) > 0. The conditional probability of E given F, defined by p(E | F), is defined as:

The events E and F are independent if and only if p(E F) = p(E)p(F)

Bernoulli Trials and Binomial Distribution

• Bernoulli Trial– Success probability p, failure probability q

The probability of exactly k successes in n independent Bernoulli trials is

Random Variables

A random variable is a function from a sample space to the real numbers

Bayes’ Theorem

Suppose that E and F are events from a sample space S such that p(E) > 0 and p(F) > 0. Then

False Positives, False Negatives

Let D be the event that a person has the disease

Let Y be the event that a person tests positive for the disease

Testing for disease

Disease is very rare: p(D) = 1/100,000

Testing is accurate:False negative: 1%False positive: 0.5%

Suppose you get a positive result, whatdo you conclude?

P(D|Y)

Spam Filtering

From: Zambia Nation Farmers Union [znfukabwe@mail.zamtel.zm]Subject: Letter of assistance for school installationTo: Richard Anderson

Dear Richard,I hope you are fine, Iam through talking to local headmen about the possible assistance of school installation. the idea is and will be welcome.I trust that you will do your best as i await for more from you.Once againThanking you very muchSebastian Mazuba.

Bayesian Spam filters

• Classification domain– Cost of false negative– Cost of false positive

• Criteria for spam– v1agra, ONE HUNDRED MILLION USD

• Basic question: given an email message, based on spam criteria, what is the probability it is spam

Email message with phrase “Account Review”

• 250 of 20000 messages known to be spam• 5 of 10000 messages known not to be spam• Assuming 50% of messages are spam, what

is the probability that a message with “Account Review” is spam

Proving Bayes’ Theorem

Expectation

The expected value of random variable X(s) on sample space S is:

Expectation examples

Number of heads when flipping a coin 3 times

Sum of two dice

Successes in n Bernoulli trials with success probability p